Expected Ntunber of Vertices.of a Random Convex Polytope
I.
Integral FOnIn.lla and Asymptotic BOlmds
by
D.G. Kelly*
Department of Statistics
Curriculum in Operations Research and Systems Analysis
Department of Mathematics
and
J.W. Tolle**
Curriculum in Operations Research and Systems Analysis
Department of Mathematics
University of North Carolina at Chapel Hill
Abstract
Given m points on the unit sphere in n-space, the hyperplanes tangent
to the sphere at the given points bound a convex polytope with m facets.
If
the points are chosen independently at random from the uniform distribution
on the sphere, the ntunber Vmn of the vertices of the polytope is a random
variable. We obtain an integral expression for EV and asymptotic bounds
mn
of the form
n (n-6)/2
a
n · Cm-n):s; EVmn
:s;
Snn(n-5)/2m •
*D.G. Kelly's research was supported by the Office of Naval Research under
Contract NOOOI4-76-C-0550.
**~.w. Tolle's research was partially supporte~ by the Office of Naval Research under
under Contract NOOOI4-76-C-0550 and partially by the Army Research Office
under Grant DAAG29-79-G-OOI4.
I.
Introduction
The main results of this working paper are:
the integral expression
(3.3) for the expected number of vertices of a random convex polytope; the
upper bound (4.1) and consequent asymptotic upper bound (4.2) on the expected
number; and the asymptotic lower bounds (4.7) and (4.8).
Briefly and infonnally, the bounds are these:
Let £1' ••• 'Em be inde-
pendent random points each having the unifonn distribution on the unit sphere
in j1 ,and let Vmn be the number of vertices of the (random) polytope whose
facets are the hyperplanes tangent to the sphere at E.J.' ••• 'Em. Then there
exist constants a and S independent of m and n such that asymptotically
The upper bound is valid for any n
~
2, asymptotically as m + 00; the lower
bound is valid for any large n, asymptotically as m + 00.
In a later paper we will present the results of numerical approximationsof the integral (3.3) for moderate values of m and n.
11.1
II.
Formulation
A.
Given an n-vector a and a real number b, the inequality aTx
b
:s;
defines a half-space in mn with bounding hyperplane H = {x: aTx =bJ. The
non-empty intersection of a finite number, say k, of such half-spaces
defines a polytope in IR n with at most k facets. A random poZytope is a
polytope generated by choosing the components of the k n-vectors, al""
,~,
and the k real munbers, .bl ' ••• ,bk , from some probability distribution on the
real line •. Obviously the structure of random polytopes will depend on the
choice of the probability distribution from which the data are drawn.
For
examples of certain distribution schemes the reader is referred to [2,4].
In this paper it is desired to consider only random polytopes in R n
with a fixed number, m, of facets.
The general scheme given above for gener-
ating polytopes will lead to redundant constraints especially in the case
where k is large relative to n.
This difficulty can be avoided if b1, ••• ,bk
are chosen as appropriate functions of the a 1 , ••• ,~.
j = l, ••• ,m, each bj satisfies
_
(2.1)
b. J
_
n
2 1/2
la·1 - ( I a.. )
i=l J1
-J
In particular, if for·
,
then the random polytope generated by the
vectors.al""'~ is
about the unit sphere, Sn-1' and has exactly m facets.
circumscribed
Each facet lies;in .
one of the hyperplanes H. = {x: a~x = la·1 J and is tangent to Slat
J
n.
~J
= a./la.
-J
-J
£.1' •• ',12m'
I•.. The
-J
-]
..
n-
polytope is thus completely.determined by the unit vectors
Consequently, an alternative generating scheme for this type of
random polytope is to choose the vectors £'l"",Em from some distribution on
Sn-l'
Of special interest here will be the case where J2J.'.",Em are independently
and uniformly distributed on Sn-l'
As is well-known ([3]) this choice of the
11.2
tangent points is equivalent to choosing the components of the vectors a.,
-J
j
= l, ••• ,m, independently from the standard normal distribution on the real
line and the b. according to (2.1).
J
.
.1t should be pointed out that requiring the polytopes to be circumscribed
on the unit sphere as described above certainly limits their possible structure.
Polytopes of certain combinatorial types cannot be realized with all of their
facets tangent to a sphere.
Therefore it may be argued that the more patholog-
ical polytopes are excluded by the method of generation •.
B.
Let
E.J.' ••• 'Em
be independent and identically distributed points on
Sn- 1. As discussed above, these points determine a unique polytope in :m. n
Let Vmn (E.!' ••• 'Em) denote the
The function Vmn is a random variable
containing Sn-l and having exactly m facets.
number of vertices of this polytope.
on the space of m-tuples of n-vectors, with values in the nonnegative integers.
The purpose of this paper is to investigate the expected value of Vmn ,
specifically when the E.j have the uniform distribution on Sn-l.
IILI
Integral Expression for EV •
mn
III.
A.
Fix integers m and n, 2 < n < m, and let £'l""'Em be independent
random points chosen front some distribution with density geE) onSn_lo
= l, ••• ,n, and let
Let Hi be the hyperplane tangent to Sn-l at ai' i
P(£'1' .. • 'En) be the polytope bounded by HI' ••• ,Hn °
Then with probability 1
any n of HI" 0,,1\ intersect in a point of lR n and there are (~) such points
of intersection, among which are the vertices of
P(£.l""'~).
Denote by A
mn the family of all n-subsets of 1,2, ... ,m.• For any
A
= {il, ••• ,in }
of H.1 , ••• ,H.1
1
n
E
mn let VA be the event that the point of intersection
A
is a vertex of P(E.:l, ••• ,n).
~
Then V is the
Stml
of the indica-
tor functions of the events VA' and so
EVmn
=
L PrcYA)
AEAmn
°
Because the E:i are identically distributed, we have by syrrnnetry
EV
mn
= (m)Pr(V
)
n
n
where Vn denotesV{l , •
B.
0
0 ,
n.} •
Now with probability 1, £.l'''''En lie on a unique small hypercircle
on Sn-1' which divides Sn-l into two caps.
of these two caps °
C(£'l,o",En)'
V
n
Let C(£'I'''.,En) be the smaller
Then Vn occurs if and only if none of
En+ l' ••• 'Em
is in
Given £'l"",En, therefore, the conditional probability of
is
So EVmn is (~) times the expected value of this function of £'1'.0. 'En; that is,
-
J
g(£)d£)m-n
C(£.I" o. ,~)
• d£.l" o. ,d~ •
(3.1)
IILZ
c.
Now we assume that the comon distribution of £1' ••• 'En is the
tmifonn distribution on Sn-l.
In this case
I _ (r)
area of C(£1' .. • 'En)
n Z
f
g (EJd£ = area of S
= '::;"zI;:"'n-_-2"7"(TI-71"l2~) ,
.
n-l
C(£1'''. 'En)
where r
=
r(£l'." 'En) is the angular radius of G(£!' .•• 'En)
(0 < r < TI/Z)
and where
Ik(r)
r . k
= f Sill
o
X
(0 < r < TI/Z, k
dx
.
= O,l,Z,.... )
is the area of a cap of angular radius r on Sk+l. Thus (3.1) can be
rewritten
EVmn
=
..
I n- Z(r) )m-n
m (
(n)E 1 - ZI
(TI/2) . .
n-Z
where the random variable r is the radius of C(£!'." ,~,).
Now it follows from results of R.A. Miles
that t
=
as],
theorem 4, p. 368)
sinZr has the beta ((n-l) Z/Z, 1/Z) distribution, whose density is
Z
proportional to t((n-l) /Z)-l(l_t)-l/Z on (0,1).
From this it follows that
the density of r is fn(n_Z)(r), where
fk(r)
= GkSinkr,
0
<
r
<
rr/Z
and
Zf(¥)
fd)fCk;l)
, k
= O,l,Z, ••••
Moreover, the distribution function of this density is
Using the above notation we obtain the expression
(3. Z)
UI.3
~/2
EVmn = (~)l
1
(1 - '! Fn-2 (r))
m-n
f n (n-2) (r)dr •
(3.3)
Although we will not use it, the following alternative expression may be of
interest.
If Bk and bk denote the distribution function and density of the
beta (k/2 , 1/2) distribution, then the substitutiont = sin 2r in (3.3) gives
IV.
Upper and Lower Bounds
A.
IV.l
The evaluation of (3.3) is not difficult in case n is Z or 3:
EVmZ
m
7T
/Z
= (Z) J
o
r m-Z Z
• - dr
7T
7T
= mel
(1 - -)
and
rIll
7T/Z
m3 = l3) J
o
EV
=
1
(1 - 7(1 -
cos r))
m-3 3 . 3 .
• 7 Sln r dr
1 m-l
Zm-4 - (2)
(m+1) (m-2) •
(We note in passing that m and m-l are the only possible values of Vm2 '
the latter occurring only when E.l, ••• ,Em are all in one semicircle. The
value of Em2V given above gives the well-known value m(~)m-l
for the
~.
probability of this event.)
The above calculations suggest that
. for fixed n, BYmn is asymptotically
of the fonn Knm. This section is devoted to finding upper and lower bounds
on BY
which establish this asymptotic growth rate and bound the growth
mn
of ~ for large n. (Unfortunately, the tempting conjecture that Kn = n-l
is false:
~ is approximately of order (/311.)n/2 as we have noted.)
B.First we will establish an upper bound by showing that for Z ~ n < m,
BYmn
~ A
onmel - Bmn )
(4.1)
where
A
-h
(~)n-l(n-l)
n(n-2) Cn- 2
n
=C
n-3
'
Before proceeding, we remark that Bmn is negligible:
C
B < (l)m-l (1 + n-2)m-l
mn
2
n-l'
C
n-Z
·
d
an slnce"""i1-T
<
3 wh en n
4"
~
3, B
mn
~
(7)m-l
1r.
IV.2
We observe also that Wallis's formula ([1], p. 258), viz.
1
r{a+ 7)
1
1
1
r(a+l) '" -(1 - 8a + 0(2))
ra
as a
a
-+
00
,
implies
cn (n-2) .'" IZTIT n
and
2.
n 2
C - '" 12rr/Cn-Z)
1
(1 - 4(n-2)) •
Thus
A '" (2rr)n/2
-n
TIe
1/4
n
(n-5)/2
and so if m ~ n, then asymptotically as n
<
BYron -
-+
00
(2rr)n/2 (n-S)/2
1/4 n
m
(4.2)
TIe
The proof of (4.1) begins with (3.3), which says that
S
ron = ~)c
n n(n-2) n-2,m-n,n(n-2)
EV
where
If
R, ~
j+1, an integration by parts using
yields
-2 1 1 k+1
2 ~-{ rr /2
1
k+1 . R,-j-1
SjkR, = r. 'k+I'(7)
+ r. + 0
r cos r dr
f (1 - 'iFj (r)) Sill
J
and hence
J
(4.3)
IV.3
2 1 1 k+1
2 ~-i
SjkJl.~-r.-F+1(2)
+'C":- + Sj,k+l,JI.-j-1
J
J
Iterating (4.5) gives, for JI.
~
forJl.~j+1.
(4.5)
v(j+1),
2 1 (1)k+1
~ ( 2)i (JI.-P(JI.-2i-1)
SjkJl. ~ - C k+1 7
i;;2
k+I)( +2]
j
(JI.-(i-1)j-(i-2)) (l)k+i
(k+i]
. '!
Cj
+ (.l.) v (JI.-j) (JI.-2i-1) ... (. t-vj - (v-I))
C.
(k+l) (K+2) ••• (k+v)
Sj ,k+v, JI.-v(j +1)
J
(4.6)
= n-2,
In particular this is valid for j
In this case for i
= 2,3, ... ,n-2
k
= m-n,
JI.
= n(n-2),
and v
= n-2.
we have
(JI.-j) (JI.-2j-1) ••• (t-(i-1)j-(i-2))
=
[(n-1) (n-2)] [(n-1) (n-3)] ••• [(n-l) (n-i)] ._ (n_l)i-1
(n-2)!
(n-1-I) !
and also by a simple integration
S
S
2
1 (1
C~)m-l).
j ,k+v,JI.-v(j+l) = n-2,m-2,n-2 = ~-2 m-1
- ,,'
Putting the two eXpressions above into (4.6) yields
S.
'.
< (__2_)n-1 (n-2)! (m-n)! (n_1)n-2
n-2,m-n,n(n-2) - C - 2
(m-I)!
n
n-1
.
_ \' C_2_)1
i=l Cn-2
L
(n-2)! (m-n)!
(n_1)i-1(~)m-n+i.
(n-1-1)!(m-n+i)J
"
Multiplying by Cn (n-2) (~) gives an upper bound on EVmn ; simplifying and
letting the sum run over j
= n-l-i
then gives (4.1).
C. Next we show that for any n
EV >
mn -
~
2 and any E
E
(0,1),
d(n-1) Z Zn-lEn
n-l .
.
en-I)
(m-n), asymptotically as m +
dn-1 n.e
I n-l
n- 1
00
•
(4.7)
IV.4
where
Before proceeding we notice that Wallis's fonnula implies .
d. ~ 12/nj
J
as j
-+-
00
•
This and Stirling's formula provide that
d (n-l) 2 2n-l £ n
(n-l) n-l
" n-l
dn-l
n-l
n.e
n-l
(2n) -r £n
~
TIe
172
n-6
-2as n
n
-+-
Therefore, for fixed large n, (4.7) implies that for any
have asymptotically as
•
£ E
(0,1), we
m-+-oo
n-l
EmnV ~
00
-r
(2TI)
TIe
1/2
n
£
6
!!.:-.
n
2
(4.8)
(m-n).
To prove (4.7) we begin again with (3.3).
For any a
E
(O,~)
(
1
(m-n...
= )Il)
In 1 - ZFn - 2 a))
~n(n-2)(a)
It can be checked that for 0 ~
a <
.z.
L · k+1
. k+3 +••• )
Fk (a ) = cos a (i
~+lsln
a + dk+3 sIna
by differentiating and using elementary properties of the gannna function.
Moreover, dO
~
dl
~ d2~ ••••
1 - 71 Fn-2 ()
a
Therefore
~
1 - cos2 a (dn-l (.
. n+la +••• ))
SInn-l a + SIn
(4.9)
IVoS
d
1Sinn-ICI.
Z cos CI.
n-
.=1 and
F ( 2)(CI.)
n n-
So, writing
T
.
~
d
(n-1)
. (n-1) 2 '
2 cos CI. S1n
CI. •
for sin CI., we have
for any
T E
(0,1).
Let E be an arbitrary number in (0,1); then-we have
EV
nm
~ (m)d
n
2
(n-1) 2
T
(n-I) E(l -
d
Z-E1 T n - 1 )m-n
if 0 < E < 1
and O· <
T
< 11-E 2
•
dn - 1 n-1
If we now write a for""""ZE"
T
,then this inequality becomes
Notice that the value of a maximizing the right side of the above inequality
d
. n-l
S of n-l < n-l(l 2) (n-l)/2
th
1S m-l.
a 1 m-l
2'E"" -E
,en
E V
nm
~
(m-n) den - 1)2
dn-1
n-l
n 1 n
2 - E (1 _ n-1l"lm-n(n_l)n-l
')
n!
m,
and (4.7) follows from this.
References
1.
Abramowitz, M., Stegtm., r. Handbook of Mathematical Functions.
Publications, New York, 1965.
2.
Dunham, J., Kelly, D., Tolle, J. Some experimental results concerning
the expected number of pivots for solving randomly generated linear
programs. Tedmical Report 77-16, ORSA Curriculum, University of
North Carolina, Chapel Hill, 1977.
3.
Feller, W. An Introduction to Probability Theory and its Applications~
vol. 2, 2nd edition. Jom Wiley & Sons, Inc., New York, 1971.
4.
Kelly, D., Tolle, J. Expected simplex algoritbm behaviour for random
linear programs. Operations Research Verfahren 31 (1979), 361-367.
5.
Miles, R. Random points, sets, and tessellations on the surface of a
sphere. Sankhya, Series A~ 33 (1971), 145-174.
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TITLE (_d Subllt/.)
Expected Number of Vertices of a Random Convex
Polytope: I. Integral Formula and Asymptotic
BOlmds
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Mimeo Series No. 1261
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SUPPLEMENTARY NOTES
...
:
19. KEY WORDS (Conllnua on 'ave,aa .Ide II naceufIt}' _d Id.nllly by blocle numbe,)
convex polytope, random polytope, vertices
20.
.
ABSTRACT (Cont/nua on ,ave,•• • Ida /I n.c••••ry _d lden/II)' b)' blocle numba,)
Given m points on the unit sphere in n-space, th~ hyperplanes tangent to the
sphere at the given points bound a convex polytope with m facets. If the
points are chosen independently at random from th,e uniform distribution on the
sphere, the number V of the vertices of the POfytope is a random variable.
nm
We obtain an integral expression for EVnm and asYmPtotic bounds of the form
exnn (n-6)/2(m-n ) ::;;EVTIm ::;; Slln (n- SD/2m .
;
•
1
!
...
I
,
I
\
i
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